Strain-Induced Topological Magnon Phase Transitions:Applications to Kagome-Lattice Ferromagnets

S. A. Owerre11Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada.

(Dated: May 1, 2018)

A common feature of topological insulators is that they are characterized by topologically invariantquantity such as the Chern number and the Z2 index. This quantity distinguishes a nontrivialtopological system from a trivial one. A topological phase transition may occur when there are twotopologically distinct phases, and it is usually defined by a gap closing point where the topologicallyinvariant quantity is ill-defined. In this paper, we show that the magnon bands in the strained(distorted) kagome-lattice ferromagnets realize an example of a topological magnon phase transitionin the realistic parameter regime of the system. When spin-orbit coupling (SOC) is neglected (i.e.no Dzyaloshinskii-Moriya interaction), we show that all three magnon branches are dispersive withno flat band, and there exists a critical point where tilted Dirac and semi-Dirac point coexist in themagnon spectra. The critical point separates two gapless magnon phases as opposed to the usualphase transition. Upon the inclusion of SOC, we realize a topological magnon phase transitionpoint at the critical strain δc = 1

2

[1 − (D/J)2

], where D and J denote the perturbative SOC and

the Heisenberg spin exchange interaction respectively. It separates two distinct topological magnonphases with different Chern numbers for δ < δc and for δ > δc. The associated anomalous thermalHall conductivity develops an abrupt change at δc, due to the divergence of the Berry curvature inmomentum space. The proposed topological magnon phase transition is experimentally feasible byapplying external perturbations such as uniaxial strain or pressure.

I. INTRODUCTION

The two-dimensional (2D) graphene sheet with negli-gible SOC is the simplest example of Dirac points (DPs),where two bands cross linearly in momentum space [1].They usually occur at the high symmetry points in theBrillouin zone (BZ), and protected by the coexistence ofinversion and time-reversal symmetry. In the distortedgraphene with unequal hopping integrals, a semi-Diracpoint (SDP) with linear band crossing along one mo-mentum direction and quadratic band crossing along theperpendicular momentum direction, can be realized byemerging two DPs [2–7]. It defines a phase transition be-tween a gapless and a gapped trivial phase. When SOCis not neglected, the SDP transforms to a gapless DP atthe topological phase transition between topological andtrivial insulator [8, 9]. As the notion of band theory isindependent of the quasiparticle excitations, the magnonbands in the 2D (distorted) honeycomb ferromagnets [10–14] directly mimic that of (distorted) graphene. To ourknowledge, however, the coexistence of (tilted) DPs andSDPs in a single system has not been studied.

Furthermore, the ideal insulating kagome-lattice fer-romagnets usually allow a SOC in the form of theDzyaloshinskii-Moriya (DM) interaction [15, 16], due tolack of an inversion center. The DM interaction generallyleads to magnonic topological insulators (mTIs) [17–20],with similar properties (such as the appearance of chiraledge modes and Chern numbers) to electronic topologicalinsulators (TIs) [21–24]. However, the experimental ob-servation of mTIs remains elusive, having been observedonly in the quasi-2D kagome ferromagnet Cu(1-3, bdc)[25], which exhibits a nonzero anomalous thermal Hall ef-fect [26]. The elusiveness of a direct experimental obser-

vation of mTIs is in part due to the fact that real kagomeferromagnetic materials may have very weak and negli-gible DM interaction as recent experiments have shownin the spin-1/2 kagome ferromagnetic mineral haydeeite,α-MgCu3(OD)6Cl2 [27]. Therefore, DPs may be intrinsicto such kagome materials with negligible effect of the DMinteraction. Moreover, most real kagome ferromagneticmaterials are imperfect and have intrinsic structural dis-tortion that deviates from the ideal structure. Besides,structural distortion can also be achieved experimentallybe applying external perturbation such as strain or pres-sure [28–31]. Hence, it is highly desirable to study howstructural distortion affects the magnon physics in Diracand topological magnetic materials.

In this paper, we present an exposition of the magnonphysics in the insulating 2D strained (distorted) kagomeferromagnets. We show that when the DM interaction isnegligible, there is no flat magnon band in the strainedkagome ferromagnets, and the DPs do not appear atthe high-symmetry points in the BZ. Interestingly, thereexists a critical distortion point where titled DPs andSDPs coexist in the magnon spectra. The critical pointseparates two gapless magnon phases with DPs. How-ever, when the DM interaction is not neglected, thereexists a topological phase transition point at δc, sepa-rating two distinct mTIs with Chern numbers (−1, 1, 0)and (−1, 0, 1). The topological magnon phase transitionpoint generates a divergent Berry curvature, which leadsto an abrupt change in the associated anomalous ther-mal Hall conductivity. The proposed topological magnonphase transition appears in the realistic parameter regimein real materials, i.e., D < J [25], where D is the per-turbative DM anisotropy, and J is the Heisenberg spinexchange interaction. Hence, we believe that the current

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topological magnon phase transition can be experimen-tally feasible in the kagome-lattice ferromagnets such asCu(1-3, bdc) with DM interactionD/J = 0.15 [25], yield-ing a strained-induced topological magnon phase transi-tion at δc = 0.48875. This can be achieved by externalperturbations such as applied uniaxial strain or pressure.

II. STRAINED KAGOME FERROMAGNETSWITHOUT DZYALOSHINSKII-MORIYA

INTERACTION

A. Spin model

We commence with the spin Hamiltonian for strained(distorted) kagome-lattice ferromagnets in the absence ofthe DM interaction,

H0 = −∑〈``′〉

J``′ ~S` · ~S`′ −B∑`

Sz` , (1)

where ~S` is the magnetic spin moment at site `, and thefirst summation runs over nearest-neighbour (NN) spins.J``′ = J on the diagonal bonds and J``′ = J ′ = Jδ on thehorizontal bonds, with δ 6= 1 as shown in Fig. 1(a). Tomodel the effect of strain, we have made the assumptionthat only the interaction along the x-axis changes, with-out lattice deformation1. The second term is an externalmagnetic field along the out-of-plane direction in unitsof gµB . There are several limiting cases of this Hamilto-nian. First, δ = 1 corresponds to an ideal kagome-latticeferromagnets. Second, δ = 0 maps to a decorated squarelattice, with additional sites at the midpoints of squarelattice edges [32]. Third, δ → ∞ maps to a quasi-1Dferromagnetic spin chain. The point δ = 0.5 or 1/2 iscritical in the magnon spectra, and separates two gaplessmagnon phases as we will show below.

B. Bosonic model

As the ground state of Eq. (1) is a fully aligned fer-romagnetic order, we describe the underlying magneticexcitations by the Holstein Primakoff (HP) transforma-tion: Sz` = S−a†`a`, S

+` ≈

√2Sa` = (S−` )†, where a†`(a`)

are the bosonic creation (annihilation) operators, andS±` = Sx` ± iS

y` denote the spin raising and lowering op-

erators. In the following, we set B = 0 as it only rescalesthe lowest magnon band at the Γ-point. The resultingnoninteracting bosonic Hamiltonian is given byHsw(~k) =

1 Alternatively, one could consider isotropic Heisenberg interac-tions with lattice deformation, i.e., only the primitive latticevectors change.

(a)

A

FIG. 1: Color online. (a) Schematic of the kagome lattice withunaxial strain along the x-axis. (b) The hexagonal BZ of thekagome lattice. Open black circles denote the locations of theDirac magnon cones in the ideal lattice J ′ = J . The red andblue dots denote the locations of the Dirac and semi-Diracmagnon cones in the strained lattice J ′ 6= J respectively.

Λ0I3×3 − Λ(~k), where Λ0 = tdiag(4, 2(1 + δ), 2(1 + δ))

Λ(~k) = 2t

0 cos k2 cos k3cos k2 0 δ cos k1cos k3 δ cos k1 0

, (2)

with t = JS and ki = ~k · ~ai. The primitive vectors aregiven by ~a1 = x, ~a2 = (x,

√3y)/2, and ~a3 = ~a2 − ~a1.

C. Dirac and Semi-Dirac magnons

For δ 6= 1, the analytical diagonalization of the Hamil-tonian Hsw(~k) is not feasible. Therefore we consider spe-cific points in the BZ. Henceforth, we label the eigen-values denoting the magnon bands as εα(~k), where α =1, 2, 3 denote the lowest, middle, and topmost magnonbands respectively. At the K-points (see Fig. 1(b)), thetwo magnon branches ε1,2(~k) are fully gapped out forδ 6= 1 with the gap given by

∆K(δ) =t

2

(√12 + 3δ(3δ − 4)− (2 + δ)

). (3)

In the ideal kagome-lattice ferromagnets, i.e., δ = 1the gap ∆K(1) = 0, which leads to magnon Dirac points(mDPs) within the magnon branches ε1,2(~k) at the high-symmetry points of the BZ, as indicated by open circlesin Fig. 1(b). There is also a quadratic magnon bandtouching point at the Γ-point within the magnon bandε2(~k) and the flat magnon band ε3(~k) for δ = 1. In con-trast, for δ 6= 1 there is no flat magnon band, and thereare mDPs in all the three magnon branches ε1,2,3(~k), aswell as magnon semi-Dirac points (mSDPs). The mDPsoccur away from the high-symmetry points in the BZ,whereas the mSDPs occur at the M-point as indicatedby red and blue dots in Fig. 1(b).

For δ 6= 1, the magnon branches ε1,2(~k) cross linearly(mDPs) at four points as opposed to a total of six for

3

Tilted mDP

mSDP

mSDP

FIG. 2: Color online. (a,b) Coexistence of tilted Dirac and semi-Dirac magnon cones in the distorted kagome ferromagnetsδ = 0.5. (c,d) Dirac magnon cones in the distorted kagome ferromagnets for δ = 0.75 and δ = 1.5 respectively. (e, f) Coexistenceof tilted mDP (pink circles) and mSDP (red circles) in the distorted kagome ferromagnets along (e) kx direction at ky = π/

√3

and along (f) ky direction at kx = 0 for δ = 0.5.

δ = 1. They are located at D1 = (±kD1x ,±π/

√3), where

kD1x = arccos

(−1 + 2δ(1− δ)− f(δ)

4δ2

), (4)

with f(δ) =√

1 + 4δ[1 + δ (2− δ(2− δ))

].

For δ > 1 there are a total of six linear magnon bandcrossings (i.e., mDPs) between ε2,3(~k) located at D2 =(±kD2

x , 0) and symmetry related points, where

kD2x = arccos

(1 + 2δ(1− δ) + f(δ)

4δ2

). (5)

For 0.5 ≤ δ ≤ 1 we obtain magnon band crossingsbetween ε2,3(~k) along the kx = 0 line located at D3 =(0,±kD3

y ), where

kD3y =

1√3

arccos(−1− 2δ + 4δ2

). (6)

The evolution of the magnon band crossing points withvarying δ is depicted in Fig. (2). Evidently, there is noflat band in the strained kagome-lattice ferromagnets forδ 6= 1. The critical point of this system where interest-ing features emerge is at δ = 0.5. As shown in Fig. 2(a),when viewed along the ky direction, the magnon branchesε1,2(~k) form linear Dirac magnon cones at D1, whereasthe magnon branches ε2,3(~k) form quadratic magnon

band touching points at M. However, in Fig. 2(b),when viewed along the kx direction, the magnon branchesε1,2(~k) form tilted linear Dirac magnon cones at D1, andthe magnon branches ε2,3(~k) form linear magnon bandtouching points at M. Therefore, tilted mDPs and mS-DPs coexist in the strained kagome-lattice ferromagneticsystems at δ = 0.5. The mSDPs result from the merg-ing of two mDPs within the magnon branches ε2,3(~k) for0.5 < δ < 1. Therefore, the coexisted mDP and mSDPsignify a phase transition between two gapless phases inthe entire system2. For the tilted mDPs at δ = 0.5, theeffective Hamiltonian near ~k = D1 can be written as

Hsw(~q + D1) = (ε0 + wqx)I2×2 + vxqxσx + vyqyσy, (7)

where σi (i = x, y, z) are Pauli matrices; ε0 is the energyof the mDPs, vx and vy are the group velocities alongqx and qy momentum direction, and w denotes the tiltparameter along the qx momentum direction. For themSDPs at δ = 0.5, the effective Hamiltonian near ~k = D3

can be written as

Hsw(~q + D3) = ε0I2×2 + vxσxqx + vyq2yσy. (8)

2 Note that although the mSDPs in the magnon branches ε2,3(~k)are gap out for δ < 0.5, mDPs exist in the magnon branchesε1,2(~k) at D1. Therefore, the entire system remains gapless.

4

FIG. 3: Color online. (Left) Evolution of the massive Dirac magnons with nonzero DM interaction D/J = 0.2. (a) δ = 0, (b)δ = 0.35, (c) δ = δc = 0.48, (d) δ = 0.75. (Right, e) The gap between ε2,3(~k) magnon branches at the M-point as a function ofδ for different values of D/J . The gap vanishes at δc and separates two mTIs with different Chern numbers (see text).

-1 0 1

kx/π

-10

0

10

-1 0 1

kx/π

-5

0

5×10

30

-1 0 1

kx/π

-10

0

10

Ω1(~k)

Ω2(~k)

Ω3(~k)

-1 0 1

kx/π

-10

0

10

-1 0 1

kx/π

-10

0

10

-1 0 1

kx/π

-10

0

10

δ = δc δ = 0.75 > δcδ = 0.35 < δc

FIG. 4: Color online. Evolution of the Berry curvature alongthe kx direction at ky = 0 (top panel) and at ky = π/

√3, i.e.,

K–M line (bottom panel) with D/J = 0.2.

III. STRAINED KAGOME FERROMAGNETSWITH DZYALOSHINSKII-MORIYA

INTERACTION

We now consider the effects of the DM interaction dueto lack of inversion center on the kagome lattice [25]. TheDM interaction is given by

HDM =∑〈``′〉

~D``′ · ~S` × ~S`′ , (9)

where ~D``′ is the DM vector between site ` and `′. Inthe linear HP spin-boson transformation, only the DMvector parallel to the magnetic field contributes to thenoninteracting bosonic Hamiltonian, but other compo-nents of the DM vector can be crucial when consideringmagnon-magnon interactions [33]. The topological as-pects of magnons can be captured clearly in the linearspin wave theory. Therefore, we consider the out-of-planeDM vector ~D``′ = Dz, which is usually present on thekagome lattice. The Fourier transformed DM interactionis given by

ΛDM(~k) = 2itD

0 − cos k2 cos k3cos k2 0 − cos k1− cos k3 cos k1 0

, (10)

with tD = DS. HenceHsw(~k) = Λ0I3×3−Λ(~k)+ΛDM(~k).

A. Topological magnon phase transition

In Fig. (3) we show that there is a topologicalphase transition at the M-point between ε2,3(~k) magnonbranches, where the bulk magnon gap closes. It occursat the critical point

δc =1

2

[1− (D/J)2

]. (11)

Indeed, a realistic topological phase transition occurswhen D < J . Interesting, this regime of the parametersalways exist in realistic kagome materials [25, 26]. Thus,we obtain two distinct topological magnon phases for δ <δc and δ > δc. They have different Chern numbers as willwe show in the subsequent subsections.

5

-1 0 1

kx/2π

0

1

2

3

4

5

6

-1 0 1

kx/2π

0

1

2

3

4

5

6

-1 0 1

kx/2π

0

1

2

3

4

5

6ǫ(k

x)/t

δ = δc δ = 0.75 > δcδ = 0.35 < δc

FIG. 5: Color online. Evolution of the bulk magnon bands(black Zebra lines) and the chiral magnon edge modes (redsolid lines ) for a strip geometry with D/J = 0.2.

Now, we would like to compare the current results toother phase transitions reported in ferromagnetic sys-tems. First, we compare our result to that of deformedgraphene with next-nearest-neighbour SOC [9], which isequivalent to a distorted honeycomb ferromagnet with anext-nearest-neighbour DM interaction [11]. In this case,the SDPs occur at δc = 2 as a single entity (i.e withoutcoexisting with DPs). It is robust against SOC (or DMinteraction for magnons), and it transforms to a DP atthe topological phase transition point at fixed δc = 2, sep-arating topological (δ < 2) and trivial (for δ > 2) insula-tor [9]. In contrast, the strained kagome-lattice ferromag-nets with DM interaction have no trivial insulator phase,besides the topological phase transition point varies withthe DM interaction as shown in Eq. (11). This is of inter-est because different kagome materials have different DMinteraction, and should in principle have different phaseboundary.

Second, let us now compare our result to that ofthe ideal (isotropic) kagome-lattice ferromagnets with aDM interaction and a second-nearest-neighbour interac-tion J2 [20], and also the XXZ kagome-lattice ferromag-nets with a DM interaction [37]. In both cases, thetopological phase boundaries occur at the K-point whenD/J =

√3 |2J2/J − 1| and D/J =

√3 respectively. Un-

fortunately, these topological phase boundaries cannot beachieved in physical materials because D > J , which isnot the case in realistic kagome-lattice materials [25, 26].

Indeed, the topological phase transition induced bystrain (lattice distortion) in Eq. (11) is evidently dif-ferent. More importantly, it is experimentally feasiblebecause a large unrealistic DM interaction is not re-quired. Therefore, we have established that the value ofthe DM interaction determines if a realistic topologicalphase transition point can be experimentally achieved.

0 0.5 1 1.5

T/J

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

κxy

0 0.5 1 1.5

δ

0.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04

0.042

κxy

δ = 0.75

δ = 0.35

δ = 0T/J = 0.9

T/J = 0.925

T/J = 0.95

FIG. 6: Color online. The anomalous thermal Hall conduc-tivity. (Left) κxy vs. δ for different values of temperature (T ).The vertical black line indicates the topological phase tran-sition point at δ = δc = 0.48 with D/J = 0.2. (Right) κxy

vs. T for different values of strain (δ) across the topologicalphase transition for D/J = 0.2. The T -dependence of κxy isdivergent at δ = δc (not shown).

B. Berry curvature

The presence of the DM interaction and the resultingmassive Dirac magnons directly imply that there shouldbe a nonzero Berry curvature. We now define the Berrycurvature of the magnon bands as

Ωα(~k) = −∑α′ 6=α

2Im[〈uα(~k)|vx|uα′(~k)〉 〈uα′(~k)|vy|uα(~k)〉

][εα(~k)− εα′(~k)

]2 ,

(12)

where vx,y = ∂Hsw(~k)/∂kx,y are the velocity operators,uα(~k) are the magnon eigenvectors. The Berry curvatureis nonzero in all regimes of δ, and its distribution in mo-mentum space is depicted in Fig. (4). It shows a verydivergent value at the topological magnon phase transi-tion point δc.

C. Chern number

To describe the topological phase transition we definethe associated Chern number as the integration of theBerry curvature over the BZ,

Cα =1

2π

∫BZ

d2~k Ωα(~k). (13)

The Chern number changes from Cα = (−1, 1, 0) in theregime δ < δc to Cα = (−1, 0, 1) in the regime δ > δc. Thesign change in the Chern number is consistent with thedistribution of the Berry curvature and the redistributionof the magnon bands. Therefore at the critical point δc,the gap between the bulk magnon bands ε2,3(~k) close, andthe resulting Chern number is ill-defined. This defines a

6

topological phase transition point separating two distinctmTIs.

Furthermore, we have solved for the chiral magnonedge modes using a strip geometry with open boundaryconditions along the y direction and infinite along x direc-tion as depicted in Fig. (5). A clear feature that emergesis the appearance of the chiral magnon edge modes inthe vicinity of the magnon bulk gaps. Although the chi-ral magnon edge modes do not cross each other for δ < δcin the ε2,3(~k) magnon branches, the entire system is stilla mTI for δ < δc, because the Chern number of the indi-vidual magnon band does not vanish identically.

D. Anomalous thermal Hall effect

One of the crucial implications of nontrivial mas-sive Dirac magnons or mTIs is that they can transportheat current in the form of the thermal Hall effect [17–19, 26, 34–36]. The thermal Hall effect can be under-stood as a consequence of the Berry curvature induced bythe DM interaction. The general formula for the trans-verse thermal Hall conductivity κxy has been derived inRef. [36]. We have used this formula to compute κxy forthe current system. The strain and temperature depen-dence of κxy are shown in Fig. 6. In the left figure, κxychanges abruptly at δ = δc (vertical black line), an in-dication of a topological magnon phase transition, whichis a direct consequence of the Berry curvature in mo-mentum space. In the right figure, we have shown theT -dependence of κxy with varying δ across the topologi-cal phase boundary. We note that the T -dependence ofκxy is divergent at δ = δc (not shown).

IV. CONCLUSION

In summary, we have studied topological magnonphase transitions in the two-dimensional strained (dis-torted) kagome-lattice ferromagnets. We showed thatin the absence of the DM interaction, Dirac and semi-Dirac points coexist in the magnon spectra, and appearbetween two distinct gapless phases with Dirac points.The inclusion of the DM interaction gapped out all the

magnon Dirac points except at the topological phasetransition point where the bulk gap closes between theoptical magnon bands, separating two distinct topologi-cal magnon insulators with different Chern numbers. Instark contrast to previous studies [20, 37], we have shownthat the strained-induced topological magnon phase tran-sitions occur in the realistic regime of the kagome-latticeferromagnets, i.e., D < J .

We also studied the anomalous thermal Hall effect andshowed that it changed abruptly at the critical topologi-cal magnon phase transition point. We believe that theseresults are of great interest in the ongoing experimentalstudies of Dirac and topological magnons in quantummagnets. Although most kagome ferromagnetic mate-rials may have intrinsic structural distortions, they canalso be tuned experimentally by applying external per-turbations such as uniaxial strain or pressure. We pro-pose to apply external strain or pressure to the cur-rently known magnon topological insulator Cu(1-3, bdc)[25], whose DM interaction is D/J = 0.15. Therefore,the current study predicts that a strained-induced topo-logical magnon phase transition should occur at δc =0.48875. In kagome ferromagnetic materials with neg-ligible DM interaction, the current study has predicteda strained-induced phase transition between two gaplessDirac points at δc = 0.5, where the phase boundaryis defined by the appearance of Dirac points coexistingwith semi-Dirac points. It will also be interesting to in-vestigate the effects of strain in the recently discovered3D Dirac magnons in Cu3TeO6 [38–40]. As topologicalmagnon phase transition has not been reported experi-mentally in real magnetic materials, we believe that thecurrent result will inspire experimentalists to manipulatethe intrinsic properties of magnetic materials using ex-ternal perturbations, which would have a great impacton the future studies of topological magnetic materials.

ACKNOWLEDGEMENTS

Research at Perimeter Institute is supported by theGovernment of Canada through Industry Canada and bythe Province of Ontario through the Ministry of Researchand Innovation.

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